AlanCrowe comments on Estimates vs. head-to-head comparisons

[AlanCrowe]

I fear that I’ve missed your point, but here is my runnable toy model written in Common Lisp

(defun x () (random 1.0))
(defun y () (random 1.0))
(defun z () (random 1.0))

(defun x-y () (- (x) (y)))
(defun y-z () (- (y) (z)))
(defun z-x () (- (z) (x)))

(defparameter diffs (list (x-y) (y-z) (z-x)))

(reduce #'+ diffs) => -0.42450535

The variable diffs get set to a list of the three estimates. Adding them up we get −0.424. What has gone wrong?

X, Y, and Z are all ¹⁄₂. But they are tricky to measure. (defun x () (random 1.0)) is modelling the idea that when we estimate X we get a random variable uniform from 0.0 to 1.0.

(defun x-y () (- (x) (y)))

is modelling the idea that we estimate X and estimate Y and subtract. (And don’t remember our estimate)

(defun y-z () (- (y) (z)))

is modelling the idea that we start from scratch, estimating Y (again) and then Z before finally subtracting.

Since the Y in X-Y is our first go at estimating Y and the Y in Y-Z is our second go at estimating Y, they have different random errors and don’t cancel like they should.